Showing papers in "International Journal of Mathematics in 2011"

Completions of monoids with applications to the cuntz semigroup

Abstract: We provide an abstract categorical framework that relates the Cuntz semigroups of the C*-algebras A and . This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C*-algebras.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WITH 𝔇⊥-PARALLEL STRUCTURE JACOBI OPERATOR

Abstract: In this paper we give some non-existence theorems for Hopf real hypersurfaces in complex two-plane Grassmannians G2(ℂm+2) with 𝔇⊥-parallel structure Jacobi operator, where 𝔇⊥ = Span {ξ1, ξ2, ξ3}.

Norm and anti-norm inequalities for positive semi-definite matrices

Abstract: Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if is a polynomial of degree m with non-negative coefficients, then, for all positive operators A, B and all symmetric norms, To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten q-norms for q ∈ (0, 1] and q < 0. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance, let f : [0, ∞) → [0, ∞) be concave and p ∈(1, ∞). If fp(t) is superadditive, then for all positive m × m matrix A = [aij]. Furthermore, for the normalized trace τ, we consider functions φ(t) and f(t) for which the functional A ↦ φ ◦ τ ◦ f(A) is convex or concave, and obtain a simple analytic criterion.

Locally compact objects in exact categories

Abstract: We identify two categories of locally compact objects on an exact category $\mathcal{A}$. They correspond to the well-known constructions of the Beilinson category $\lim\limits_{\leftrightarrow}\,\mathcal{A}$ and the Kato category $\kappa(\mathcal{A})$. We study their mutual relations and compare the two constructions. We prove that $\lim\limits_{\leftrightarrow}\,\mathcal{A}$ is an exact category, which gives to this category a very convenient feature when dealing with K-theoretical invariants, and study the exact structure of the category $\lim\limits_{\leftrightarrow}\, {\bf Vect}_0(k)$ of Tate spaces. It is natural therefore to consider the Beilinson category $\lim\limits_{\leftrightarrow}\,\mathcal{A}$ as the most convenient candidate to the role of the category of locally compact objects over an exact category. We also show that the categories ${\rm Ind}_{\aleph_0}(\mathcal{C})$, ${\rm Pro}_{\aleph_0}(\mathcal{C})$ of countably indexed ind/pro-objects over any category $\mathcal{C}$ can be described...

Representations of surface groups in the projective general linear group

Abstract: Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let $\mathcal{R}_G$ be the moduli space of reductive representations of π1X in G. We determine the number of connected components of $\mathcal{R}_{{\rm PGL}(n,\mathbb{R})}$, for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in $\mathcal{R}_{{\rm SL}(3,\mathbb{R})}$ is homotopically equivalent to $\mathcal{R}_{{\rm SO}(3)}$.

On the structure of some reduced amalgamated free product c*-algebras

Abstract: We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

Obstructed bundles of rank two on a quintic surface

Abstract: In this note we consider the moduli space of stable bundles of rank two on a very general quintic surface. We study the potentially obstructed points of the moduli space via the spectral covering of a twisted endomorphism. This analysis leads to generically non-reduced components of the moduli space, and components which are generically smooth of more than the expected dimension. We obtain a sharp bound asked for by O'Grady saying when the moduli space is good.

Dirac structures of omni-lie algebroids

Abstract: Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid 𝔇E ⊕ 𝔍E is necessarily a Lie algebroid together with a representation on E. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in ${\mathcal{T}} = TM \oplus E$; we establish the relation between the normalizer NL of a reducible Dirac structure L and the derivation algebra Der(b (L)) of the projective Lie algebroid b(L); we study the cohomology group H•(L, ρL) and the relation between NL and H1(L, ρL); we describe Lie bialgebroids using the adjoint representation; we study the deformation of a Dirac structure L, which is related with H2(L, ρL).

Fine gradings on exceptional simple lie superalgebras

Abstract: The fine abelian group gradings on the simple exceptional classical Lie superalgebras over algebraically closed fields of characteristic 0 are determined up to equivalence.

The Cuntz semigroup of continuous functions into certain simple C*-algebras

Abstract: This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C0(X, A), where A is a unital, simple, -stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray–von Neumann semigroups of C(K, A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(𝕋, A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

On second cohomology of duals of compact groups

Abstract: We show that for any compact connected group G the second cohomology group dened by unitary invariant 2-cocycles on ^ G is canonically isomorphic to H 2 ( \ Z(G);T). This implies that the group of autoequivalences of the C -tensor category RepG is isomorphic to H 2 ( \(G);T) o Out(G). We also show that a compact connected group G is completely determined by RepG. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classication of triangular semisimple Hopf algebras. In two appendices we give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analogue of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

Signature pairs for group-invariant hermitian polynomials

Abstract: We study the signature pair for certain group-invariant Hermitian polynomials arising in CR geometry. In particular, we determine the signature pair for the finite subgroups of SU(2). We introduce the asymptotic positivity ratio and compute it for cyclic subgroups of U(2). We calculate the signature pair for dihedral subgroups of U(2).

On the stability of syzygy bundles

Abstract: We are concerned with the problem of the stability of the syzygy bundles associated to base-point-free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from M. Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if its rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner [Comment. Math. Helv.59 (1984) 635–650]. We then give an elementary proof of H. Brenner's criterion of stability for monomial syzygy bundles, avoiding the use of Klyachko's results on toric vector bundles. We finally prove the existence of stable syzygy bundles defined by monomials of the same degree d, of any possible rank, for n at least 3. This extends the similar result proved, for n = 2, by L. Costa, P. Macias Marques and R. M. Miro-Roig [J. Pure Appl. Algebra214 (2010) 1241–1262]. The extension to the case n at least 3 has been also, independently, obtained by P. Macias Marques in his thesis [arXiv:0909.4646/math.AG (2009)].

Smooth projective symmetric varieties with picard number one

Abstract: We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover, we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth.

Approximation of weakly singular plurisubharmonic functions

Abstract: In this paper we give some properties of the weighted energy class eχ and study the approximation in these classes. We prove that any function in eχ can be approximated by an increasing sequence of plurisubharmonic functions defined on larger domains and with finite χ-energy.

Elliptic genera of complete intersections in weighted projective spaces

Abstract: We generalize the residue method of Ma–Zhou to compute the two-variable elliptic genera of smooth hypersurfaces and complete intersections in weighted projective spaces.

Classification of graded left-symmetric algebraic structures on witt and virasoro algebras

Abstract: We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

Schematic harder–narasimhan stratification

Abstract: For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder–Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder–Narasimhan stratification. In this paper, we show how to endow each Harder–Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback family admits a relative Harder–Narasimhan filtration with a given Harder–Narasimhan type if and only if the base change factors through the schematic stratum corresponding to that Harder–Narasimhan type. The above schematic stratification induces a stacky stratification on the algebraic stack of pure-dimensional coherent sheaves. We deduce that coherent sheaves of a fixed Harder–Narasimhan type form an algebraic stack in the sense of Artin.

MINIMAL DYNAMICS AND $\mathcal{Z}$-STABLE CLASSIFICATION

Abstract: Let X be an infinite compact metric space, α : X → X a minimal homeomorphism, u the unitary that implements α in the transformation group C*-algebra C(X) ⋊α ℤ, and a class of separable nuclear C*-algebras that contains all unital hereditary C*-subalgebras of C*-algebras in . Motivated by the success of tracial approximation by finite dimensional C*-algebras as an abstract characterization of classifiable C*-algebras and the idea that classification results for C*-algebras tensored with UHF algebras can be used to derive classification results up to tensoring with the Jiang-Su algebra , we prove that (C(X) ⋊α ℤ) ⊗ Mq∞ is tracially approximately if there exists a y ∈ X such that the C*-subalgebra (C*(C(X), uC0(X\{y}))) ⊗ Mq∞ is tracially approximately . If the class consists of finite dimensional C*-algebras, this can be used to deduce classification up to tensoring with for C*-algebras associated to minimal dynamical systems where projections separate tracial states. This is done without making any assumptions on the real rank or stable rank of either C(X) ⋊α ℤ or C*(C(X), uC0(X\{y})), nor on the dimension of X. The result is a key step in the classification of C*-algebras associated to uniquely ergodic minimal dynamical systems by their ordered K-groups. It also sets the stage to provide further classification results for those C*-algebras of minimal dynamical systems where projections do not necessarily separate traces.

Stable bundles of rank 2 with four sections

Abstract: This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

ON THE TOPOLOGY OF POLYNOMIAL MAPPINGS FROM ℂn TO ℂn-1

Abstract: We consider a polynomial map from ℂn to ℂn-1 and prove that if there exists a so-called very good projection with respect to the value t0, then this value is an atypical value for the map if and only if the Euler characteristic of the fibers are not constant. We describe some topology of the fibers and prove that there is no extension of the characterization of the atypical value via the Lojasiewicz number as in the case n = 2.

KdV GEOMETRIC FLOWS ON KÄHLER MANIFOLDS

Abstract: In this paper, we define a kind of KdV (Korteweg–de Vries) geometric flow for maps from a real line ℝ or a circle S1 into a Kahler manifold (N, J, h) with complex structure J and metric h as the generalization of the vortex filament dynamics from a real line or a circle. By Hasimoto transformation, we find that the KdV geometric flow on a Riemann surface of constant Gauss curvature is just classical complex-valued mKdV equation. From the view point of geometric analysis we show that the Cauchy problems of KdV flow on a Kahler manifold admits a unique local solution in suitable Sobolev spaces. In the case the target manifold (N, J, h) with complex structure J and metric h is a certain type of locally Hermitian symmetric space, we show that the KdV flow exists globally by exploiting the conservation laws and semi-conservation law of KdV flow.

Connection on parabolic vector bundles over curves

Abstract: Let E* be a parabolic vector bundle over a smooth complex projective curve. We prove that E* admits an algebraic connection if and only if the parabolic degree of every parabolic vector bundle which is a direct summand of E* is zero. In particular, all parabolic semistable vector bundles of parabolic degree zero admit an algebraic connection.

The relative weak asymptotic homomorphism property for inclusions of finite von neumann algebras

Abstract: A triple of finite von Neumann algebras B ⊆ N ⊆ M is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators {uλ}λ∈Λ in B such that for all x,y ∈ M. We prove that a triple of finite von Neumann algebras B ⊆ N ⊆ M has the relative weak asymptotic homomorphism property if and only if N contains the set of all x ∈ M such that for a finite number of elements x1, …, xn in M. Such an x is called a one-sided quasi-normalizer of B, and the von Neumann algebra generated by all one-sided quasi-normalizers of B is called the one-sided quasi-normalizer algebra of B. We characterize one-sided quasi-normalizer algebras for inclusions of group von Neumann algebras and use this to show that one-sided quasi-normalizer algebras and quasi-normalizer algebras are not equal in general. We also give some applications to inclusions L(H) ⊆ L(G) arising from containments of groups. For example, when L(H) is a masa we determine the unitary normalizer algebra as the von Neumann algebra generated by the normalizers of H in G.

GEOMETRIC KdV FLOWS, MOTIONS OF CURVES AND THE THIRD-ORDER SYSTEM OF THE AKNS HIERARCHY

Abstract: By applying the concept of geometric KdV flows into Kahler and para-Kahler manifolds, we give a unified geometric interpretation for the four typical integrable equations in the third-order system of the AKNS hierarchy. That is: the KdV equation, the mKdV equation, the complex mKdV- equation and the complex mKdV+ equation are, respectively, equivalent to the first kind reduction, the second kind reduction of the geometric KdV flow of maps from R1 to the de Sitter two-space S1,1 ↪ R2,1 (regarded as a para-Kahler manifold), the geometric KdV flow of maps from R1 to the hyperbolic two-space H2 ↪ R2,1 (regarded as a Kahler manifold of noncompact type) and the geometric KdV flow of maps from R1 to the two-sphere S2 ↪ R3 (regarded as a Kahler manifold of compact type). This is an application of the general geometric KdV flows.

Homotopy decompositions of gauge groups over riemann surfaces and applications to moduli spaces

Abstract: For a prime p, the gauge group of a principal U(p)-bundle over a compact, orientable Riemann surface is decomposed up to homotopy as a product of spaces, each of which is commonly known This is used to deduce explicit computations of the homotopy groups of the moduli space of stable vector bundles through a range, answering a question of Daskalopoulos and Uhlenbeck

An Extension of the Cartan Nochka- Second Main Theorem for Hypersurfaces

Abstract: In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in ℂPn relative to a possibly degenerate set of hyperplanes. In this paper, we generalize Nochka's theorem to the case of curves in a complex projective variety intersecting hypersurfaces in subgeneral position. Further work will be needed to determine the optimal notion of subgeneral position under which this result can hold, and to lower the effective truncation level which we achieved.

A classification of smooth embeddings of four-manifolds in seven-space, ii

Abstract: Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0 Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12 This group acts on E7(N) by an embedded connected sum Boechat and Haefliger constructed an invariant ℵ: E7(N) → H2(N;ℤ) which is injective on the orbit space of this action; they also described im(ℵ) We determine the orbits of the action: for u ∈ im(ℵ) the number of elements in ℵ-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2 The proof is based on Kreck's modified formulation of surgery

Noncommutative mackey theorem

Abstract: In this note we investigate quantizations of the weak topology associated with a pair of dual linear spaces We prove that the weak topology admits only one quantization called the weak quantum topology, and that weakly matrix bounded sets are precisely the min-bounded sets with respect to any polynormed topology compatible with the given duality The technique of this paper allows us to obtain an operator space proof of the noncommutative bipolar theorem

Some results on the non-riemannian quantity h of a finsler metric

Abstract: In this paper, we study the non-Riemannian quantity H in Finsler geometry. We obtain some rigidity theorems of a compact Finsler manifold under some conditions related to H. We also prove that the S-curvature for a Randers metric is almost isotropic if and only if H almost vanishes. In particular, S-curvature is isotropic if and only if H = 0.